Two-dimensional problem

The heat transfer problems discussed in the preceding subsection are steady-state convection-diffusion problems with the general variable varying in one dimension only. We now turn our attention to the unsteady state two-dimensional convection-diffusion problem which includes a source term S. The problem is described by

Similar to the case of one-dimensional convection-diffusion, different discretization schemes for the discretized equations (7) – (14) can be obtained by using different expressions for A(|PeΔ|) from the following table.

Table Summary of A(|PeΔ|) for different schemes

Scheme

A(|PeΔ|)

Central difference

Upwind

1

Hybrid

Exponential

Power Law

Three-dimensional problem

The discretized equation for a transient three-dimensional convection-diffusion problem can be obtained by integrating the conservation equation with respect to t in the interval of (t, t+Δt) and over the three-dimensional control volume P (formed by considering two additional neighbors at top, T, and bottom, B). The final form of the governing equation is [1]

The Different discretization schemes for the above three-dimensional problem can be obtained by using different expressions for A(|PeΔ|) from Table 4.3.
In addition to the six first order discretization schemes described above, some researchers have used higher order schemes such as second order upwind (Leonard et al., 1981) and QUICK (Quadratic Upwind Interpolation of Convective Kinetics; Leonard, 1979) schemes to overcome the false diffusion problem, which is referred to as error caused by using the discretization scheme with accuracy less than the second order (Patankar, 1980). The error due to false diffusion could potentially be severe for (1) transient problems, (2) multidimensional steady-state problems, or (3) problems with non-constant source terms (Tao, 2001). While the accuracies of these higher order schemes are better than the first order schemes, their computational time is much greater than that of the first order schemes.

References

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